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Spring Semester 2021

Date / Time Speaker Title Location
17 February 2021
15:00-16:15 		Dr. Ignacio Barros
Orsay
Event Details

Algebraic Geometry and Moduli Seminar

Title On the irrationality of moduli spaces of K3 surfaces
Speaker, Affiliation Dr. Ignacio Barros, Orsay
Date, Time 17 February 2021, 15:00-16:15
Location Zoom
Abstract I will report on recent joint work with D. Agostini and K.-W. Lai, where study how the degrees of irrationality of the moduli space of polarized K3 surfaces grow with respect to the genus g. We prove that, for a series of infinitely many genera, the irrationality is bounded by the Fourier coefficients of certain modular forms of weight 11, and thus grow at most polynomially, in terms of g. Our proof relies on results of Borcherds on Heegner divisors together with results of Hassett and Debarre--Iliev--Manivel on special cubic fourfolds and Gushel--Mukai fourfolds.
On the irrationality of moduli spaces of K3 surfacesread_more
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19 February 2021
16:00-17:15 		Dr. Carl Lian
HU Berlin
Event Details

Algebraic Geometry and Moduli Seminar

Title Non-tautological and H-tautological Hurwitz cycles
Speaker, Affiliation Dr. Carl Lian, HU Berlin
Date, Time 19 February 2021, 16:00-17:15
Location Zoom
Abstract We will explain the construction of a large family of new non-tautological algebraic cycles on moduli spaces of curves coming from Hurwitz spaces. Namely, the locus of stable curves of sufficiently large genus admitting a degree d cover of a curve of genus h>0 is non-tautological, with appropriate marked points added and subject to the non-vanishing of the d-th Fourier coefficient a certain modular form. This builds on examples of Graber-Pandharipande and van Zelm in the case (d,h)=(2,1). Time-permitting, we will discuss how these cycles fit instead into a larger theory of "H-tautological" classes on moduli spaces of admissible Galois covers of curves.
Non-tautological and H-tautological Hurwitz cyclesread_more
Zoom
24 February 2021
15:00-16:15 		Prof. Dr. Jenia Tevelev
UMass Amherst
Event Details

Algebraic Geometry and Moduli Seminar

Title Scattering amplitudes of stable curves
Speaker, Affiliation Prof. Dr. Jenia Tevelev, UMass Amherst
Date, Time 24 February 2021, 15:00-16:15
Location Zoom
Abstract Equations of hypertree divisors on the Grothendieck-Knudsen moduli space of stable rational curves, introduced by Castravet and Tevelev, appear as numerators of scattering amplitude forms for n massless particles in N=4 Yang-Mills theory in the work of Arkani-Hamed, Bourjaily, Cachazo, Postnikov and Trnka. We re-interpret and generalize MHV scattering amplitude forms as probabilistic Brill-Noether theory: the study of statistics of images of n marked points on a Riemann surface under a random meromorphic function. This leads to a beautiful physics-inspired geometry for various classes of algebraic curves: smooth, stable, hyperelliptic, real algebraic, etc.
Scattering amplitudes of stable curvesread_more
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26 February 2021
16:00-17:15 		Miguel Moreira
ETH Zürich
Event Details

Algebraic Geometry and Moduli Seminar

Title A rationality result on the GW theory of local Hirzebruch surfaces
Speaker, Affiliation Miguel Moreira, ETH Zürich
Date, Time 26 February 2021, 16:00-17:15
Location Zoom
Abstract In this talk I’ll report on joint work in progress with Tim Buelles. Our work concerns the enumerative geometry of local Hirzebruch surfaces. The main result we prove says that, for a fixed genus and curve class β, the generating function obtained from Gromov-Witten invariants in such genus and curve class β+iF (where F is the fiber class) is a rational function, and moreover it satisfies a Q ⟷ Q^{-1} symmetry.
A rationality result on the GW theory of local Hirzebruch surfacesread_more
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3 March 2021
15:00-16:15 		Dr. Sam Molcho
ETH Zürich
Event Details

Algebraic Geometry and Moduli Seminar

Title The logarithmic tautological ring
Speaker, Affiliation Dr. Sam Molcho, ETH Zürich
Date, Time 3 March 2021, 15:00-16:15
Location Zoom
Abstract Given a pair of a smooth variety X with a normal crossing divisor D, I will discuss the construction of a subring of the Chow ring of X, called the logarithmic tautological ring, generated by certain "tautological" classes obtained from the strata of D. I will explain the basic structure of the logarithmic tautological ring: its behavior under blowups, its relation to combinatorics, and a method to compute it. I will conclude by relating the logarithmic tautological ring of the moduli space of curves with the double ramification cycle, and explain how the structure of the logarithmic tautological ring implies our recent result with R.Pandharipande and J.Schmitt that the double ramification cycle is a product of divisors in a blowup of \M_{g,n}.
The logarithmic tautological ringread_more
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* 5 March 2021
17:00-18:15 		Prof. Dr. Emily Clader
San Francisco State University
Event Details

Algebraic Geometry and Moduli Seminar

Title Permutohedral complexes and curves with cyclic action
Speaker, Affiliation Prof. Dr. Emily Clader, San Francisco State University
Date, Time 5 March 2021, 17:00-18:15
Location Zoom
Abstract Although the moduli space of genus-zero curves is not toric, it shares some of the combinatorial structure that a toric variety would enjoy. In fact, by adjusting the moduli problem slightly, one finds a moduli space that is indeed toric, known as Losev-Manin space. The associated polytope is the permutohedron, which also encodes the group-theoretic structure of the symmetric group. Batyrev and Blume generalized this story by constructing a "type-B" version of Losev-Manin space, whose associated polytope is a signed permutohedron that relates to the group of signed permutations. In joint work with C. Damiolini, D. Huang, S. Li, and R. Ramadas, we carry out the next stage of generalization, defining a family of moduli space of curves with Z_r action encoded by an associated "permutohedral complex" for a more general complex reflection group, which specializes when r=2 to Batyrev and Blume's moduli space.
Permutohedral complexes and curves with cyclic actionread_more
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10 March 2021
15:00-16:15 		Arkadij Bojko
Oxford University
Event Details

Algebraic Geometry and Moduli Seminar

Title Wall-crossing for Hilbert schemes on CY 4-folds I
Speaker, Affiliation Arkadij Bojko, Oxford University
Date, Time 10 March 2021, 15:00-16:15
Location Zoom
Abstract Moduli schemes of coherent sheaves on CY 4-folds have a natural 3-term obstruction theory which leads to a definition of virtual fundamental classes. To study these, Joyce proposed a conjectural wall-crossing framework. We review the construction of vertex algebras which give rise to wall-crossing formulae of the virtual fundamental classes in the homology of the moduli stack of sheaves. We define a vertex algebra on the moduli stack of the auxiliary category of JS pairs and motivate their wall-crossing formula. Unlike the standard Behrend--Fantechi construction, the VFC depend on an additional piece of data called orientation. We review the recent results of existence of orientations and their compatibility under direct sums. They give rise to signs satisfying co-cycle conditions, necessary to define these vertex algebras.
Wall-crossing for Hilbert schemes on CY 4-folds Iread_more
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12 March 2021
16:00-17:15 		Arkadij Bojko
Oxford University
Event Details

Algebraic Geometry and Moduli Seminar

Title Wall-crossing for Hilbert schemes on CY 4-folds II
Speaker, Affiliation Arkadij Bojko, Oxford University
Date, Time 12 March 2021, 16:00-17:15
Location Zoom
Abstract We use the wall-crossing conjecture of the previous talk to study Hilbert schemes of points. We translate it into a fully explicit and computable problem and use it to address a conjecture of Cao--Kool for the generating series of tautological integrals for line bundles. We compute virtual fundamental classes of 0-dimensional sheaves for fixed orientations and wall-cross back to get closed expressions for generating series of VFC of Hilbert schemes, which allow us to compute Segre series, Verlinde series and Nekrasov's genus. We show that a large family of invariants can be related via a universal transformation to invariants on elliptic surfaces. At the end we discuss some other (potential) applications.
Wall-crossing for Hilbert schemes on CY 4-folds IIread_more
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17 March 2021
15:00-16:15 		Prof. Dr. Martijn Kool
University of Utrecht
Event Details

Algebraic Geometry and Moduli Seminar

Title Proof of Magnificent Four
Speaker, Affiliation Prof. Dr. Martijn Kool, University of Utrecht
Date, Time 17 March 2021, 15:00-16:15
Location Zoom
Abstract Motivated by super-Yang–Mills theory on a Calabi–Yau 4-fold, Nekrasov and Piazzalunga assigned weights to r-tuples of solid partitions (4-dimensional piles of boxes) and conjectured a formula for their weighted generating function. We define K-theoretic virtual invariants of Quot schemes of 0-dimensional quotients of O^r on affine 4-space by realizing them as zero loci of isotropic sections of orthogonal bundles. Using the Oh–Thomas localization formula, we recover Nekrasov–Piazzalunga’s weights. Applying ideas from Okounkov in the 3-dimensional case, we prove Nekrasov-Piazzalunga’s formula. Joint work with J. Rennemo.
Proof of Magnificent Fourread_more
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19 March 2021
16:00-17:15 		Dr. Woonam Lim
ETH Zürich
Event Details

Algebraic Geometry and Moduli Seminar

Title Virtual invariants of Quot schemes of surfaces I
Speaker, Affiliation Dr. Woonam Lim, ETH Zürich
Date, Time 19 March 2021, 16:00-17:15
Location Zoom
Abstract Let S be a smooth projective surface. Grothendieck’s Quot scheme on S admits a 2-term perfect obstruction theory when it parametrizes torsion quotients. This allows us to define various virtual invariants of Quot schemes, including homological (K-theoretic) descendent invariants and virtual Segre/Verlinde numbers. The study of these invariants is partially motivated by the parallel theory of moduli of sheaves. A special feature of the Quot scheme theory is the conjectural rationality of the homological (K-theoretic) descendent series. We explain how this can be proven for all surfaces with p_g>0 using the multiplicative structural formula involving Seiberg-Witten invariants. The same method applies to the study of the virtual Segre/Verlinde series. We also explain the virtual Segre/Verlinde correspondence and a special symmetry for punctual Quot schemes which is reminiscent of the numerical strange duality.
Virtual invariants of Quot schemes of surfaces Iread_more
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24 March 2021
15:00-16:15 		Dr. Woonam Lim
ETH Zürich
Event Details

Algebraic Geometry and Moduli Seminar

Title Virtual invariants of Quot schemes of surfaces II
Speaker, Affiliation Dr. Woonam Lim, ETH Zürich
Date, Time 24 March 2021, 15:00-16:15
Location Zoom
Abstract Let S be a smooth projective surface. Grothendieck’s Quot scheme on S admits a 2-term perfect obstruction theory when it parametrizes torsion quotients. This allows us to define various virtual invariants of Quot schemes, including homological (K-theoretic) descendent invariants and virtual Segre/Verlinde numbers. The study of these invariants is partially motivated by the parallel theory of moduli of sheaves. A special feature of the Quot scheme theory is the conjectural rationality of the homological (K-theoretic) descendent series. We explain how this can be proven for all surfaces with p_g>0 using the multiplicative structural formula involving Seiberg-Witten invariants. The same method applies to the study of the virtual Segre/Verlinde series. We also explain the virtual Segre/Verlinde correspondence and a special symmetry for punctual Quot schemes which is reminiscent of the numerical strange duality.
Virtual invariants of Quot schemes of surfaces IIread_more
Zoom
26 March 2021
16:00-17:15 		Dr. Irene Schwarz
ETH Zürich
Event Details

Algebraic Geometry and Moduli Seminar

Title On the Kodaira dimension of the moduli space of hyperelliptic curves with marked points
Speaker, Affiliation Dr. Irene Schwarz, ETH Zürich
Date, Time 26 March 2021, 16:00-17:15
Location Zoom
Abstract It is known that the moduli space H_{g,n} of genus $g$ stable hyperelliptic curves with $n$ marked points is uniruled for n <= 4g+5. We consider the complementary case and show that H_{g,n} has non-negative Kodaira dimension for n = 4g+6 and is of general type for n >= 4g+7. Important parts of our proof are the calculation of the canonical divisor and establishing that the singularities do not impose adjunction conditions.
On the Kodaira dimension of the moduli space of hyperelliptic curves with marked pointsread_more
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31 March 2021
15:00-16:15 		Dr. Woonam Lim
ETH Zürich
Event Details

Algebraic Geometry and Moduli Seminar

Title Virtual invariants of Quot schemes of surfaces III
Speaker, Affiliation Dr. Woonam Lim, ETH Zürich
Date, Time 31 March 2021, 15:00-16:15
Location Zoom
Abstract Let S be a smooth projective surface. Grothendieck’s Quot scheme on S admits a 2-term perfect obstruction theory when it parametrizes torsion quotients. This allows us to define various virtual invariants of Quot schemes, including homological (K-theoretic) descendent invariants and virtual Segre/Verlinde numbers. The study of these invariants is partially motivated by the parallel theory of moduli of sheaves. A special feature of the Quot scheme theory is the conjectural rationality of the homological (K-theoretic) descendent series. We explain how this can be proven for all surfaces with p_g>0 using the multiplicative structural formula involving Seiberg-Witten invariants. The same method applies to the study of the virtual Segre/Verlinde series. We also explain the virtual Segre/Verlinde correspondence and a special symmetry for punctual Quot schemes which is reminiscent of the numerical strange duality.
Virtual invariants of Quot schemes of surfaces IIIread_more
Zoom
2 April 2021
16:00-17:15 		Prof. Dr. Ravi Vakil
Stanford
Event Details

Algebraic Geometry and Moduli Seminar

Title Vector bundles on the projective line and Bott periodicity
Speaker, Affiliation Prof. Dr. Ravi Vakil, Stanford
Date, Time 2 April 2021, 16:00-17:15
Location Zoom
Abstract H. Larson recently completely described (integrally) the "characteristic classes" of vector bundles on P^1-bundles, in the Chow ring. Bott periodicity relates vector bundles on a topological space X to vector bundles on X x S^2: the "moduli space" BU of complex vector bundles is "basically the same as" the "moduli space" of maps of a (pointed) sphere to BU. I will try to explain the algebro-geometric incarnation of Bott periodicity. (This is work in progress with H. Larson.)
Vector bundles on the projective line and Bott periodicityread_more
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9 April 2021
16:00-17:15 		Prof. Dr. Jérémy Guéré
Université de Grenoble Alpes
Event Details

Algebraic Geometry and Moduli Seminar

Title Gromov-Witten theory of a quotient of the quintic threefold
Speaker, Affiliation Prof. Dr. Jérémy Guéré, Université de Grenoble Alpes
Date, Time 9 April 2021, 16:00-17:15
Location Zoom
Abstract Let X be a smooth quintic hypersurface in P^4 defined by a chain polynomial and G be its maximal cyclic group of symmetry, which has order 256. In this talk, I will describe a general method in order to compute via localization the Gromov-Witten theory of the orbifold quotient [X/G] in all genera and all degrees. I will explain various ideas coming from my paper on Hodge-Gromov-Witten theory and from Fan-Lee's paper on quantum Lefschetz. I will also give at the end an interesting attempt (which unfortunately contains a mistake at the moment) at computing quintic GW invariants using Costello's theorem.
Gromov-Witten theory of a quotient of the quintic threefoldread_more
Zoom
23 April 2021
16:00-17:15 		Dr. Elden Elmanto
Harvard University
Event Details

Algebraic Geometry and Moduli Seminar

Title A sales pitch of (A^1)-homotopy theory to geometers
Speaker, Affiliation Dr. Elden Elmanto, Harvard University
Date, Time 23 April 2021, 16:00-17:15
Location Zoom
Abstract I will give a "user's guide" to (unstable) A^1-homotopy theory in the sense of Morel and Voevodsky, aimed at geometers. I will then give two recent geometric applications of the theory - a reformulation of the purity conjectures of Colliot-Thélène and Sansuc (with Kulkarni and Wendt) and an improvement of the splitting results of Bhatwadekar, Das and Mandal for vector bundles over real schemes (with Asok).
A sales pitch of (A^1)-homotopy theory to geometersread_more
Zoom
28 April 2021
15:00-16:15 		Prof. Dr. Y.-P. Lee
University of Utah
Event Details

Algebraic Geometry and Moduli Seminar

Title Quantum K-theory I
Speaker, Affiliation Prof. Dr. Y.-P. Lee, University of Utah
Date, Time 28 April 2021, 15:00-16:15
Location Zoom
Abstract I will give an introductory talk on the basics of quantum K-theory, assuming only rudimentary Gromov-Witten theory.
Quantum K-theory Iread_more
Zoom
30 April 2021
16:00-17:15 		Prof. Dr. Georg Oberdieck
Universität Bonn
Event Details

Algebraic Geometry and Moduli Seminar

Title A multiple cover rule for the Hilbert scheme of a K3 surface
Speaker, Affiliation Prof. Dr. Georg Oberdieck, Universität Bonn
Date, Time 30 April 2021, 16:00-17:15
Location Zoom
Abstract It was conjectured that all Gromov-Witten invariants of K3 surfaces in imprimitive curve classes are determined by those in primitive classes. In this talk I will explain a parallel conjecture for the class of hyperkahler varieties which are deformation equivalent to the Hilbert scheme of points of a K3 surface. This case is more subtle.The description of the monodromy by Markman plays an important role. Using Noether-Lefschetz theory for the Fano variety of lines of a cubic fourfold I will give some evidence for this conjecture for K3[2]-type. I also give applications to the Noether-Lefschetz theory of Debarre-Voisin fourfolds.
A multiple cover rule for the Hilbert scheme of a K3 surfaceread_more
Zoom
5 May 2021
15:00-16:15 		Prof. Dr. Y.-P. Lee
University of Utah
Event Details

Algebraic Geometry and Moduli Seminar

Title Quantum K-theory II
Speaker, Affiliation Prof. Dr. Y.-P. Lee, University of Utah
Date, Time 5 May 2021, 15:00-16:15
Location Zoom
Abstract Further development of the K-theory of the moduli space of stable maps.
Quantum K-theory IIread_more
Zoom
7 May 2021
16:00-17:15 		Prof. Dr. Ben Bakker
University of Illinois at Chicago
Event Details

Algebraic Geometry and Moduli Seminar

Title Algebraic approximation of Calabi-Yau varieties and the decomposition theorem
Speaker, Affiliation Prof. Dr. Ben Bakker, University of Illinois at Chicago
Date, Time 7 May 2021, 16:00-17:15
Location Zoom
Abstract Calabi-Yau manifolds are built out of simple pieces by the Beauville–Bogomolov decomposition theorem: up to an etale cover, any Kahler manifold with torsion first Chern class is a product of complex tori, irreducible holomorphic symplectic manifolds, and strict Calabi-Yau manifolds (which have no holomorphic forms except a holomorphic volume form). Work of Druel-Guenancia-Greb-Horing-Kebekus-Peternell over the last decade has culminated in a generalization of this result to projective Calabi-Yau varieties with the kinds of singularities that arise in the MMP, though the proofs critically use algebraic methods. In this talk I will describe joint work with H. Guenancia and C. Lehn on the deformation theory of Calabi-Yau varieties. Among other things, this allows us to extend the decomposition theorem to Kahler varieties and also resolves the K-trivial case of a conjecture of Peternell asserting that any minimal Kahler variety can be approximated by algebraic varieties.
Algebraic approximation of Calabi-Yau varieties and the decomposition theoremread_more
Zoom
12 May 2021
15:00-16:15 		Dr. Dhruv Ranganathan
Cambridge University
Event Details

Algebraic Geometry and Moduli Seminar

Title Logarithmic DT theory
Speaker, Affiliation Dr. Dhruv Ranganathan, Cambridge University
Date, Time 12 May 2021, 15:00-16:15
Location Zoom
Abstract Logarithmic DT theory is a framework for sheaf counting on a simple normal crossings threefold pair. It is meant to be the “sheaf side” of the curve/sheaf correspondence in this logarithmic universe. I will explain how the sheaf theory side seems to require a fundamentally different attack than the ones used in the development of logarithmic GW theory. I’ll then try to outline where this theory should sit in the larger context of virtual enumerative geometry, and how the logarithmic degeneration formulas interact with these conjectures.
Logarithmic DT theoryread_more
Zoom
14 May 2021
16:00-17:15 		Prof. Dr. Paolo Rossi
Università di Padua
Event Details

Algebraic Geometry and Moduli Seminar

Title Double ramification cycles, meromorphic differentials and (2+1)-dimensional integrable hierarchies
Speaker, Affiliation Prof. Dr. Paolo Rossi, Università di Padua
Date, Time 14 May 2021, 16:00-17:15
Location Zoom
Abstract Both the double ramification cycles and the closure of the spaces of meromorphic differentials with zero residues form partial CohFTs whose phase space is infinite dimensional. The double ramification hierarchy construction works fine for such generalizations of CohFTs. In two recent papers with A. Buryak we proved that in the first case one gets a (2+1)-dimensional version of KdV on a noncommutative torus and that our DR/DZ equivalence conjecture (the DR hierarchy and the Dubrovin-Zhang hierarchy are equivalent) implies that the generating function of intersection numbers of a DR cycle with any monomial in the psi classes satisfies this hierarchy. I will then mention a work in progress where, with similar techniques, we show that the DR hierarchy associated to the closure of the spaces of meromorphic differentials with zero residues contains the KP hierarchy as a reduction corresponding to differentials with exactly 2 zeros. Once more this can be applied to the computation of intersection numbers of the space of such differentials with monomials in psi classes.
Double ramification cycles, meromorphic differentials and (2+1)-dimensional integrable hierarchiesread_more
Zoom
19 May 2021
15:00-16:15 		Prof. Dr. Vivek Shende
UC Berkeley
Event Details

Algebraic Geometry and Moduli Seminar

Title Boundaries, skeins, and recursion
Speaker, Affiliation Prof. Dr. Vivek Shende, UC Berkeley
Date, Time 19 May 2021, 15:00-16:15
Location Zoom
Abstract I will define open Gromov-Witten invariants for Calabi-Yau 3-folds in all genera. The key idea is to count curves by their boundary in the skein modules of Lagrangians. Then I will prove the following conjecture of Ooguri and Vafa: the colored HOMFLYPT polynomials of a knot are counts of holomorphic curves in the resolved conifold with boundary on a Lagrangian associated to the knot. In the process we will see the geometric origin of recursion relations for colored knot invariants. This talk presents joint work with Tobias Ekholm.
Boundaries, skeins, and recursionread_more
Zoom
* 21 May 2021
17:00-18:00 		Hannah Larson
Stanford University
Event Details

Algebraic Geometry and Moduli Seminar

Title Intersection theory on low-degree Hurwitz spaces
Speaker, Affiliation Hannah Larson, Stanford University
Date, Time 21 May 2021, 17:00-18:00
Location Zoom
Abstract While there is much work and conjectures surrounding the intersection theory of the moduli space of curves, relatively little is known about the intersection theory of the Hurwitz space H_{k, g} parametrizing smooth degree k, genus g covers of P^1. For k = 3, 4, 5, I will present a stabilization result for the rational Chow rings of H_{k,g} as g tends to infinity. In the case k = 3, we completely determine the Chow ring. As a corollary, we prove that the Chow groups of the simply branched Hurwitz space are zero in codimension up to roughly g/k for k = 3, 4, 5. This is joint work with Sam Canning.
Intersection theory on low-degree Hurwitz spacesread_more
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* 21 May 2021
18:15-19:15 		Samir Canning
UC San Diego
Event Details

Algebraic Geometry and Moduli Seminar

Title The Chow rings of the moduli spaces of curves of genus 7, 8, and 9
Speaker, Affiliation Samir Canning, UC San Diego
Date, Time 21 May 2021, 18:15-19:15
Location Zoom
Abstract I will explain how to build upon the results in Hannah Larson's talk on the intersection theory on low-degree Hurwitz spaces in order to compute the Chow rings of the moduli spaces of curves of genus 7, 8, and 9. I'll focus specifically on tetragonal curves and on hexagonal curves in genus 9, using work of Mukai in the latter case. This is joint work with Hannah Larson.
The Chow rings of the moduli spaces of curves of genus 7, 8, and 9read_more
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28 May 2021
16:00-17:15 		Dr. Maria Yakerson
ETH Zürich
Event Details

Algebraic Geometry and Moduli Seminar

Title Hermitian K-theory via oriented Gorenstein algebras
Speaker, Affiliation Dr. Maria Yakerson, ETH Zürich
Date, Time 28 May 2021, 16:00-17:15
Location Zoom
Abstract Hermitian K-theory is a younger cousin of algebraic K-theory, it classifies vector bundles with quadratic forms. In this talk, we will discuss a new geometric model for hermitian K-theory as a motivic space, given by the Hilbert scheme of Gorenstein subschemes of infinite affine space that are equipped with an orientation. This is joint work with Marc Hoyois, Joachim Jelisiejew and Denis Nardin.
Hermitian K-theory via oriented Gorenstein algebrasread_more
Zoom
* 4 June 2021
18:00-19:15 		Prof. Dr. Jim Bryan
UBC
Event Details

Algebraic Geometry and Moduli Seminar

Title Counting invariant curves on a Calabi-Yau threefold with an involution
Speaker, Affiliation Prof. Dr. Jim Bryan, UBC
Date, Time 4 June 2021, 18:00-19:15
Location Zoom
Abstract Gopakumar-Vafa invariants are integers n_beta(g) which give a virtual count of genus g curves in the class beta on a Calabi-Yau threefold. In this talk, I will give a general overview of two of the sheaf-theoretic approaches to defining these invariants: via stable pairs a la Pandharipande-Thomas (PT) and via perverse sheaves a la Maulik-Toda (MT). I will then outline a parallel theory of Gopakumar-Vafa invariants for a Calabi-Yau threefold X with an involution. They are integers n_beta(g,h) which give a virtual count of curves of genus g in the class beta which are invariant under the involution and whose quotient by the involution has genus h. I will give two definitions of n_beta(g,h) which are conjectured to be equivalent, one in terms of a version of PT theory, and one in terms of a version of MT theory. The invariants can be computed and the conjecture proved in the case where X=AxC where A is an Abelian surface with the involution given by multiplication by -1. In this case, the formulas for the invariants are given in terms of Jacobi forms and the specialization of n_beta(g,h) to h=0 recovers the count of hyperelliptic curves on A first computed by B-Oberdieck-Pandharipande-Yin. This is joint work with Stephen Pietromonaco.
Counting invariant curves on a Calabi-Yau threefold with an involutionread_more
Zoom
* 18 June 2021
14:00-15:15 		Younghan Bae
ETH Zürich
Event Details

Algebraic Geometry and Moduli Seminar

Title Counting surfaces on Calabi-Yau 4-folds
Speaker, Affiliation Younghan Bae, ETH Zürich
Date, Time 18 June 2021, 14:00-15:15
Location HG G 19.1
Abstract After the work of Klemm and Pandharipande in 2007, counting curves on Calabi-Yau fourfold has been an important subject of the enumerative geometry. Unlike counting curves, not much is known for counting surfaces. In this talk we will consider various sheaf theoretic invariants which count surfaces on Calabi-Yau 4-folds. I will give examples which suggest new phenomena in the 4-fold theory. Some consequences will follow from conjectural correspondences between those theories. This is work in progress with Martijn Kool and Hyeonjun Park.
Counting surfaces on Calabi-Yau 4-foldsread_more
HG G 19.1

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